34;- WELL
Chapter 6: Conclusions This study has achieved its main aims of
Chapter 6: Conclusions
radial grids gIven the difference In the structure of the underlying differential equations.
In comparing analytical solutions to the results of finite-difference simulation some discrepancies were noted. These could be traced to the use of a constant flux boundary condition at the well bore in some analytical solutions as opposed to the more appropriate constant pressure boundary condition used by the simulation.
Analytical solutions that used a constant pressure boundary condition proved to be in good agreement with the simulation results. The importance of this finding is two- fold:
1. It demonstrates convergence between the analytical and simulation models as the degree of spatial and temporal discretization is increased, which shows that this approach is sufficient to model the problem
2. It highlights the fact that engineers need to be careful when trying to match early-time data using the analytical methods available in many of the commercial well test analysis packages
In some cases the phenomenon of well bore storage would mask the early time data but in cases using downhole shut-in valves these data would be preserved. Although the differences are small, conclusions regarding the degree of permeability anisotropy based on matching the early time data with inappropriate analytical methods could be invalid. This would have consequences for the estimated recovery factor from an oil field and might influence the decision to develop a field or optimum choice of development scenario for a field. Many of the analytical techniques used for analysing horizontal well tests are also based on constant flux assumptions and can be expected to suffer from the same problems at early time.
Application of the simulation approach to a real well test of a partially penetrating well showed the power of the technique in being able to model lateral and vertical property changes (absolute permeability, saturation and hence effective fluid mobility, reservoir topography, no-flow boundaries). These variations in properties could not be modelled in combination using purely analytical methods and proved to be important in understanding the pressure response at the well in question. Ignoring factors that cannot be modelled analytically could easily result in erroneous interpretations. For example:
• Tilting of the reservoir into the aquifer resulted in a steadily decreasing derivative that could easily be mistaken for an increase in permeability at a distance from the well
• The contrast in permeability over the perforated and unperforated intervals magnified the apparent skin implying that only a small mechanical skin was necessary to match the test
While both these influences on the pressure behaviour were suspected prior to running the simulations, it was only through the simulation that these could be confirmed.
Thus, the simulation provided a basis for testing hypotheses regarding the nature of the pressure behaviour.
In the case of the well being studied, it was possible to demonstrate that the vertical permeability must be high in order to match the early time pressure behaviour. This implies that the vertical sweep efficiency (ability to uniformly replace oil by water) of this reservoir is also likely to be high. Oil recovery can therefore be expected to be towards the upper side of the range of previously forecasted values with delayed water breakthrough at the wells. Both of these factors (i.e. higher total oil recovery and faster oil recovery) considerably enhance the value of the field in net-present-value (NPV) terms. Recent production data from the field have confirmed that recoveries are beyond expectation.
The use of numerical simulation to model well test behaviour is a powerful technique that should be applied more often. It is of particular relevance to the high cost offshore environment where development decisions can involve hundreds of millions, or in extreme cases billions, of dollars. These decisions often need to be taken on the basis of very limited data. Although numerical simulation of well test pressure behaviour is extremely time consuming in comparison to the use of purely analytical techniques it has the potential to extract additional information regarding reservoir properties from the limited data available. This in turn should lead to a better understanding of the risks involved in developing the field concerned and a more accurate expectation value for the field concerned.
References
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2. Muskat, M., "Use of Data on the Build-up of Bottomhole Pressures", Trans.
AIME (1937) 123,44-48.
3. Miller,
c.c.,
Dyes, AB., and Hutchinson, C.A, "The Estimation of Permeability and Reservoir Pressure from Bottom Hole Pressure Build-up Characteristics", Trans. AIME (1950) 189,91-104.4. Homer, D.R., "Pressure Build-Up in Wells", Proc. Third World Pet. Cong., EJ.Brill, Leiden (1951) 11, 503.
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10. Gringarten, A
c.,
and Ramey, H. 1., "An Approximate Infinite Conductivity Solution for a Partially Penetrating Line Source Well", SPEJ, (April 1975), 140.11 Bilhartz, H. L., and Ramey, H. 1., "The Combined Effects of Storage, Skin, and Partial Penetration on Well Test Analysis", SPE paper 6753, presented at the SPE 52nd Annual Technical Conference and Exhibition, Denver, (October
1977)
12. Saidikowski, R. M., "Numerical Simulations of the Combined Effects of Wellbore Damage and Partial Penetration", SPE paper 8204, presented at the
SPE 54th Annual Technical Conference and Exhibition, Las Vegas, (September 1979).
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14. Strauss, J. P., "Simulation Studies of the _ Area-", SOEKOR internal report SOE-PEN-RPT-049 (April 1992).
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17. van Everdingen, AF., "The Skin Effect and its Impediment to Fluid Flow into a Wellbore", Trans. AIME(1953), 198,171-176.
18. Odeh, A S., and Babu, B. K., Transient Flow Behaviour of Horizontal Wells, Pressure Drawdown and Build-up Analysis", SPE paper 18802 (1989).
19. Ozkan, E., and Raghavan, R., "New Solutions for Well-Test-Analysis Problems: Part 1- Analytical Considerations; Part 2 - Computational Considerations and Applications", SPEFE. (September 1991),359-378.
20. Ding, W., and Reynolds, A
c.,
"Computation of the Psuedoskin Factor for a Restricted-Entry Well", SPEFE, (March 1994),9-14.21. Stre1stova-Adams, T. D., "Pressure Drawdown in a Well with Limited Flow Entry", JPT, (November 1979),1469-1476.
22. Strelstova-Adams, T. D., "Pressure Transient Analysis for Afterflow- Dominated Wells Producing from a Reservoir with Gas Cap", JPT, (April.
1981), 172-174.
23. Al-Khalifa, A 1., and Odeh, A S., "Well Test Analysis in Oil Reservoirs with Gas Caps and/or Water Aquifers", SPE paper 19842, presented at the SPE 64th Annual Technical Conference and Exhibition, San Antonio, (October 1989).
24. Earlougher, R. C., Jr., "Advances III Well Test Analysis", SPE of AIME Monograph Series (1977) 5.
25. Dake, L. P., "Fundamentals of Reservoir Engineering", Elsevier Scientific Publishing Company (1978).
26. Abdou, M. K., Pham, H. D., and Al-Aqueeli, A. S., "Use of Orthogonal and Nonorthogonal Grids for the Simulation of a Faulted Reservoir", SPE paper 21391, presented at the 1991 SPE Middle East Oil Show, Bahrain, (November.
1991)
27. Heinemann, Z. E., and Deimbacher, F. X., "Advances in Reservoir Simulation Gridding", Proc. Fourth Int1. Forum on Reservoir Simulation (September 1992)
28. Amando, L. C. N., Ganzer, L., and Heinemann, Z. E., "Finite Volume Discretization of the Fluid Flow Equations on General Perpendicular Bisection Grids", Proc. Fifth Int1. Forum on Reservoir Simulation (December 1994) 29. Aziz, K., and Settari, A., "Petroleum Reservoir Simulation", Elsevier Applied
Science Publishers (1979)
30. Aziz, K., "Fundamentals of Reservoir Simulation", Course notes prepared for Stanford University, California (1993)
31. Intera, "Eclipse 100 - Technical Appendices (Release 94A)", Intera Information Technologies Ltd, Oxfordshire (1993)
32. Vinsome, P.L.W., "Orthomin, an Iterative Method for Sparse Banded Sets of Simultaneous Equations", SPE paper 5729, presented at the SPE-AIME Fourth Symposium on Reservoir Simulation, Los Angeles, (1976)
33. Appleyard, J. R., and Cheshire, 1. M., "Nested Factorization", SPE paper 12264, presented at the Seventh SPE Symposium on Reservoir Simulation, San Francisco, (1983)
34. Cheshire, 1. M., "The Solution of Linear Equations in Implicit Simulators", Proc. Fourth Int1. Forum on Reservoir Simulation (September 1992)
35. Holmes, 1. A., "Enhancements to the Strongly Coupled, Fully Implicit Well Model: Wellbore Crossflow Modeling and Collective Well Control", SPE
paper 12259, presented at the Seventh SPE Symposium on Reservoir Simulation, San Francisco, (1983)
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Edwards Inc., (1946)
37. Kuchuk, F. 1., Goode, P. A., Wilkinson, D. 1., and Thambynayagam, R. K. M.,
"Pressure Transient Behaviour of Horizontal Wells With and Without Gas Cap or Aquifer", SPE paper 17413, presented at the SPE California Regional Meeting in Long Beach, California, (1988)
38. Ozkan, E., Sarica,
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Haciislamoglu, M., and Raghavan, R., "Supplement to SPE 24683. Effect of Conductivity on Horizontal-Well Pressure Behaviour", SPE paper 30230, (1995)39. Ozkan, E., Sarica, C., Haciislamoglu, M., and Raghavan, R., "Effect of Conductivity on Horizontal-Well Pressure Behaviour", SPE paper 24683, SPE Adv. Tech. Series, Vol. 3, No. 1, (1995)
40. Yiidiz, T., and Bassiouni, Z., "Transient Pressure Analysis in Partially Penetrating Wells", SPE Paper 21551 presented at Joint CIM/SPE International Technical Meeting, Calgary, (June 1990)
41. Stehfest, H, "Algorithm 368 - Numerical Inversion of Laplace Transforms", Comm. ACM, Vol. 13, No. 1, (January 1970)
42. Talbot, "The Accurate Numerical Inversion of Laplace Transforms", J. Inst.
Math. Appl., Vol. 23, (1979)
43. Yanosik, 1. L., and McKraken, T. A., "A Nine-Point, Finite-Difference Reservoir Simulator for Realistic Prediction of Adverse Mobility Ratio Displacements", SPEJ, (August 1979)
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46 Winters, S. and Pfderkamfer, H., "Geological and Geophysical Appraisal of the _ Field''', SOEKOR Internal Report, 1994
47 Winters, S., "Architectural Geometries of Fan 5 in the Skoorsteenberg Region, Tanqua Sub-basin, Permian Ecca, South Africa", SOEKOR Internal Report SOE-EXP-RPT-0327, 1995
48 Burger, C. A. 1., Van Niekerk, A. B., Ridley, T. P., and Strauss, 1. P., " _ Development Study*", SOEKOR Internal Report SOE-PET-RPT-152, 1994 49 Burger, C. A.1., Personal Communication, 2002
*
Name ofwell or field withheldfor reasons ofconfidentiality.Roman Letters:
A The Jacobian matrix for solution on the finite difference equations (Sections 3.6 to 3.10 only).
Am Surface area of the m-th face of the grid-block being considered.
b Represents the residual in the Newton iterations used in solving the finite difference equations (Sections 3.6 to 3.10 only).
bp Penetration ratio in formulae for partial penetration skin.
B An easily inverted approximation to the Jacobian matrix used in solved the finite difference equations (Sections 3.6 to 3.10 only).
Bo ' Bw ' Bg Formation volume factors for oil, water, and gas respectively.
C Total isothermal compressibility.
Co, cw 'cg Isothermal compressibilities for oil, water, and gas respectively.
Cr Compressibility associated with the reduction of pore space as pressure Increases.
d
D
F
g k
Diagonal In the tri-diagonal Jacobian matrix (Sections 3.6 to 3.10 only).
Depth.
Vector representing the flow terms in the black-oil fluid flow equations (Section 3.2, Equation 51).
Acceleration due to gravity.
Permeability.
Component of permeability in the horizontal and vertical directions respectively.
Horizontal permeability over the perforated interval.
km' krw, k,t: Relative permeability to oil, water, and gas respectively.
Ko, K1 Modified Bessel functions of the second kind of order 0 and 1 respectively.
h
1
m M
p
Pwl
P p
q Q
Formation thickness.
Length of well interval that is open to flow.
Lengths defining the completion geometry (Figure lA).
Hydrostatic head correction for calculating well flow (Section 3.2, Equation 84).
The identity matrix.
Lower side bands in the Jacobian matrix (Sections 3.6 to 3.10 only).
Slope from a Homer plot.
Mass accumulation vector in black-oil fluid flow formula (Section 3.2, Equation 50).
Pressure.
Initial pressure, i.e. pressure before any flow has taken place to disturb initial equilibrium.
Well bottom hole pressure, finite difference model (Section 304).
Capillary pressure.
Well flowing pressure.
Laplace transform of pressure.
Matrix used in the nested factorisation process in solving a set of linear equations (Section 3.9).
Fluid flow rate.
Mass sink vector associated with a well 111 the finite difference equations (Section 3A).
r
r111\1
Radius or radial distance from centre of well to a given position.
Residual vectors in solution of a set of linear equations (Sections 3.8 to 3.10 only).
Saturation normalised residual vectors in solution of a set of linear equations (Sections 3.8 to 3.10 only).
Radius of investigation.
Wellbore radius.
6J.ro, 6J.rl, 6J.rn Residual search directions m solution of a set of linear equations (Sections 3.8 to 3.10 only).
R
s
S
Sill
T
T
u
Radial distance to face of adjacent grid cell (Section 4.3, Figure 4.4).
Radial distance between adjacent grid points (Section 4.5).
Residual vector for the solution of the non-linear finite difference equations (Section 3.2).
Radii used in calculation oftransmissibilities (Section 3.3).
The transformed equivalent to time in the Laplace transform for pressure.
The skin factor.
The fluid saturations for oil, water, and gas respectively.
Flow convergence or partial penetration skin.
Mechanical skin (skin due to near well damage).
Time.
Temperature when discussing PVT properties or transmissibility when discussing flow from grid-block to grid-block.
Matrix used in nested factorisation procedure when solving a set of linear equations (Section 3.9 only).
Flow velocity as a scalar property.
u
v
w
x,y
x
Upper side bands in the Jacobian matrix (Sections 3.6 to 3.10 only).
Flow velocity as a vector property.
Volume.
Represents the well state, i.e. flowing pressure and produced fluid fractions for water and gas (Section 3.4).
Horizontal position in a Cartesian reference system.
The change in the state vector at each non-linear iteration when solving the finite difference equations, note that this is equivalent to the solution of the linear equations (Sections 3.6 to 3.10 only).
Estimates of required solution for set of linear equations (Sections 3.8 to 3.10 only).
&0' &1' &n Solution search directions for solution of a set of linear equations (Sections 3.8 to 3.10 only).
x
z
z·
State vector for finite difference model, i.e. contains the value of oil pressure, water saturation, and gas saturation for every grid block (Section 3.2).
Vertical position (Cartesian or cylindrical reference system).
Vertical distance between adjacent grid points (Section4.5).
Intercept on a Homer plot, i.e. extrapolation of best-fit line to infinite time.
Greek Letters:
a o, ai' an Weighting factors used in solution of finite difference equations to minimise residual when moving along a particular search direction.
f3
Weighting factor, used in nine point method to determine optimum contribution of diagonal grid points (Section 4.5 only).c, p
Y
()
Subscripts:
D
i, j, k
Weighting factors, used in solution of finite difference equations to ensure each search direction is orthogonal to the last direction (Section 3.8 only).
Implies a change or difference.
Time discretisation error.
Density.
Fluid viscosity.
Fluid viscosity across the perforated interval.
Fluid potential.
Fluid mobility (Section 3.2).
Porosity.
Hydrostatic pressure gradient for a fluid, i.e. product of fluid density and acceleration due to gravity.
Elder's constant, i.e. Ye = 0.5772,..
Diagonal matrix used in solution of set of linear equations with tri- diagonal structure (Sections 3.7 to 3.10 only).
Homer or superposition/transformed time.
Pressure potential.
The exponent ofEuler's constant, i.e. ~
=
exp(0.5772,..).Tangential component in a cylindrical co-ordinate system.
Implies a dimensionless property.
Typically used as indices, when used in combination in a grid system they represent the grid block index according to the three co-ordinate axes.
o,w,g x,y,z v, h p
Oil, water, or gas phase respectively.
Directions/components in a Cartesian reference system.
Vertical and horizontal direction/component respectively.
Either phase (oil, water, or gas) or partial depending on context.
Anisotropic - Implies properties that differ in value depending on the direction they are measured in.
Absolute Permeability - The permeability associated with a porous medium when the pore space is completely filled by a single fluid phase. (See also permeability, effective permeability, and relative permeability)
Build-up - A period, following a flow period, when the well is closed to prevent further flow to surface. During this time the pressure in the well will steadily increase (build-up).
Capillary Pressure - The pressure difference between immiscible fluid phases that share the pore space. This pressure difference is a consequence of the interfacial tension between the phases.
Claystone - A rock that contains a large proportion of clay minerals and other very fine grained material.
Compressibility - A measure of the degree to which the volume of a substance can be changed by the application of pressure, i.e. the ratio of volume change to pressure change per unit volume.
Coning - The formation of a cone of fluid from an overlying or underlying zone to the well in response to the pressure drop at the well with the typical consequence that unwanted fluid (water or gas) is produced.
Darcy Units - A hybrid system of units that avoids the use of unnecessary conversion factors in flow equations and has units of more convenient size than the S.l. (Table 1.1: Comparison of Different Unit Systems)
Drainage - A dynamic process where the saturation of the wetting phase (i.e. the phase that preferentially wets the surfaces of the grains) decreases with time.
Drill stem test (DST) - A well test that is conducted using drill pipe to convey the fluid from the reservoir to surface or vice versa. (See also Well Test)
Effective Permeability - The permeability associated with a particular fluid phase when the pore space is filled by more than one fluid phase. (See also permeability, and relative permeability)
Field Units - A system of imperial units that is commonplace in the oil industry (Table 1.1)
Formation Volume Factor - The ratio of reservoir volume to surface volume for a given fluid.
Heterogeneous - Implies properties that change from one position to another.
History Matching - The process of adjusting the properties in the simulation model so that it is able to closely reproduce the observed production history.
Homogenous - Implies properties that are the same everywhere throughout the system (note that a body may have homogenous but anisotropic properties).
Horner Plot - A specialized plot for analysing pressure behaviour during a build-up.
The x-axis is based on a transformed time function that is designed to give a straight line on the plot when transient flow (i.e. infinitely acting radial flow) is taking place.
The slope of the line is inversely proportional to the permeability thickness of the interval contributing to flow (Sections 1.3 and 5.4).
Imbibition - A dynamic process where the saturation of the wetting phase (i.e. the phase that preferentially wets the surfaces of the grains) increases with time.
Interbedded - Alternating layers of different rock types.
Isotropic - Implies properties that are identical in all directions.
Lithology - Rock type, typically referring to the general characteristics of the rock such as bulk composition and texture, for example.
Massive - As applied to describing rocks, it implies a single continuous body without significant internal structure.
Net Present Value (NPV) - A measure of the total value of a future cash flow that takes the time value of money (i.e. money now is better than money tomorrow) into account. This is achieved by discounting cash flows by progressively larger amounts the further in the future they occur. NPV can therefore be described as a cumulative discounted cash flow.
Partially penetrating well - A well that does not completely penetrate the reservoir interval (Figure 1.2a).
Partially completed well - A well where only part of the reservoir interval is open to flow (Figure 1.2b).
PEBI Grid - A perpendicular bisection grid, also known as a Voronoi grid. This is a flexible gridding technique where the faces of the grid blocks perpendicularly bisect the lines joining adjacent grid blocks (Section 3.1).
Permeability - The fluid conductivity of a porous medium, i.e. a measure of the ease with which fluid can move through the medium. Permeability is a directional property and is therefore normally described by a diagonal property tensor. It is defined through Darcy's Law (Section 1.2, Equations 1 to 4).
Pore space - Space between the solid particles (grains) that make up the rock.
Porosity - The fraction of the bulk volume of rock that is not solid (pore space) and is therefore available for occupation by gas or liquids.
PVT - Refers to measurements taken under changing conditions of pressure, volume, and temperature (Section 2.6).
Relative Permeability - A factor that is applied to absolute permeability in order to account for the reduction of the effective permeability when part of the pore space is filled by another fluid (Section 2.2, Equation 21).
Sandstone - A rock that comprises mainly relatively coarse grains
Cl
/16 to 2 mm in size), typically made up predominantly of quartz.Saturation - The volume fraction of the pore space that is filled by a particular fluid.
Siltstone - A fine-grained rock with a grain size less than sandstone and coarser than shale/claystone.
Seismic - Refers to the elastic waves that are used in order to assess sub-surface geometry and properties. The waves are generated by a source and the resulting reflections recorded. Changes in the elastic properties will cause the waves to reflect and/or refract and thus interpretation of the recordings allow reconstruction of what lies below the surface.
Skin - A factor that is introduced to account for an additional (or reduced) pressure drop over and above that expected from homogenous transient flow (Section 1.3). It is a manifestation of the difference between near-well and larger scale properties. In